Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere

2023/04/06 by Michelen, Marcus, Yakir, Oren
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2304.02898

Abstract

Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of n points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltrán and Shub considered the set of points given by the stereographic projection of the roots of the random elliptic polynomial of degree n and computed the expected logarithmic energy. We study the fluctuations of the logarithmic energy associated to this random configuration and prove a central limit theorem. Our approach shows that all cumulants of the logarithmic energy are asymptotically linear in n, and hence the energy is well-concentrated on the scale of √(n).

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